Full Chapter Overview

Variable

A symbol or placeholder that represents an unknown or general value, utilized in mathematical expressions and equations. It can represent numbers, categories, or both, depending on the context.

Set

A well-defined collection of distinct objects, known as elements or members, in which the order of the elements is irrelevant. Sets are often denoted by curly braces, e.g., {1, 2, 3}.

Universal Statement

A type of assertion that claims a property or condition holds true for all elements within a specified set. For example, 'All positive numbers are greater than zero' is a universal statement about the set of positive numbers.

Conditional Statement

A logical expression that describes a relationship between two statements using an 'if-then' format. It conveys that whenever the first statement (antecedent) is true, the second statement (consequent) will also be true. For instance, 'If a number is divisible by 18, then it is divisible by 6.'

Existential Statement

A declaration that asserts the existence of at least one element within a particular set that satisfies a certain property or condition. For instance, 'There exists a prime number that is even' states there is at least one even prime number.

Subset (⊆)

A relation that indicates that all elements of Set A are also contained within Set B, without requiring that A and B be identical. A proper subset is a special case where Set A contains some but not all elements of Set B.

Proper Subset (⊂)

A designation indicating that Set A is a subset of Set B, and in addition, Set A does not contain all the elements found in Set B, meaning that the two sets are not equivalent.

Union (∪)

An operation that results in a new set containing all distinct elements from both Set A and Set B, essentially combining the two sets without duplication.

Intersection (∩)

An operation that produces a set consisting of elements that are common to both Set A and Set B, effectively identifying shared values between the two sets.

Difference (A - B)

An operation that yields a set containing elements that belong to Set A but not to Set B, representing the remaining members of Set A after removing those in Set B.

Cartesian Product (A × B)

A mathematical operation that forms a new set containing all possible ordered pairs (a, b) where 'a' is an element from Set A and 'b' is an element from Set B. This operation accounts for every combination of elements from the two sets.

Function

A specific type of relation between two sets where each input from Set A is associated with exactly one output from Set B, ensuring that no input corresponds to multiple outputs.

Graph

An abstract mathematical structure consisting of vertices (also known as nodes) connected by edges, which may represent relationships, connections, or pathways between the nodes.

Directed Graph

A graph in which edges have a specific direction, typically represented by arrows, indicating the relationship flows from one vertex to another. This directional aspect is important for understanding the nature of relationships represented in the graph.

Undirected Graph

A graph where the edges do not have a specified direction, meaning the connection between vertices is mutual, and the relationship is bidirectional.

Relation

A set of ordered pairs that express a connection or relationship between two sets. Each pair contains an element from each set, indicating how members of the two sets are associated.

Function Machine

A conceptual model used to visualize a function as a process that takes an input, applies a predetermined rule or operation, and produces a unique output, akin to a black box that transforms inputs into outputs.

Ordered Pair

A pair of elements (a, b) in which the arrangement matters, meaning that (a, b) is distinct from (b, a). Ordered pairs are fundamental in defining relations and Cartesian products.

Elements (Members)

The individual objects or values that constitute a set. Each element is considered a distinct entity within the context of the set.

Domain

The complete set of all possible first elements (inputs) in a relation or function, signifying the values that can be fed into the function.

Codomain

The set that includes all possible outputs for a function, defining the range of outputs that could theoretically occur, but not all outputs in the codomain need to be realized.

Range

The subset of the codomain that consists of all actual outputs generated by a function when applied to the inputs from its domain.

Real Numbers (R)

The set comprising all rational and irrational numbers, encompassing values on the continuous number line which include integers, fractions, and non-repeating, non-terminating decimals.

Integers (Z)

The complete set of whole numbers that includes all positive and negative numbers as well as zero, represented as {..., -3, -2, -1, 0, 1, 2, 3, ...}.

Rational Numbers (Q)

Numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero, allowing for values that can be represented as finite or repeating decimals.

Natural Numbers (N)

The set of positive integers typically used for counting (1, 2, 3, ...), with some definitions including zero (0, 1, 2, 3, ...).

Set-Roster Notation

A way of defining a set by explicitly listing each of its elements within curly braces, such as {1, 2, 3}.

Set-Builder Notation

A method of defining a set by specifying a property that its members must satisfy, usually expressed in the form {x | condition on x}.

Empty Set (∅ or {})

A set that contains no elements at all, representing the concept of 'nothingness' in set theory.

Cardinality

The measure of the number of distinct elements present in a set, which can be finite or infinite.

Power Set

The collection of all possible subsets of a given set, including the empty set and the set itself, denoted as P(S) for a set S.

Injective (One-to-One) Function

A type of function where each distinct input corresponds to a distinct output, ensuring that no two different inputs map to the same output.

Surjective (Onto) Function

A function where every element in the codomain has at least one corresponding input in the domain, meaning the range of the function covers the entire codomain.

Bijective Function

A function that is both injective and surjective, establishing a one-to-one correlation between all elements in the domain and codomain.

Vertex (Node)

A fundamental part of a graph, representing a point where two or more edges meet.

Edge

A connection between two vertices in a graph, representing a relationship or pathway.

Degree (of a vertex)

The total number of edges that are incident to a vertex, indicating how many connections it has.

Path

A sequence of vertices where each adjacent pair is connected by an edge, effectively outlining a route within the graph.

Cycle

A particular type of path in a graph that begins and ends at the same vertex, creating a closed loop.